L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s + 2·13-s + 16-s − 2·17-s + 2·20-s + 22-s − 25-s − 2·26-s + 2·29-s − 4·31-s − 32-s + 2·34-s − 2·37-s − 2·40-s − 10·41-s + 4·43-s − 44-s + 4·47-s + 50-s + 2·52-s + 2·53-s − 2·55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.213·22-s − 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.316·40-s − 1.56·41-s + 0.609·43-s − 0.150·44-s + 0.583·47-s + 0.141·50-s + 0.277·52-s + 0.274·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.31428750094090372811842113001, −6.74373972124873346554358089110, −5.98771079406763610733343708056, −5.53618566531056577967176687482, −4.63545890380665990141670986613, −3.68670878539381830801305511269, −2.80378494620370130161308447703, −2.00850567963734584766683280983, −1.30473186878917396922066338559, 0,
1.30473186878917396922066338559, 2.00850567963734584766683280983, 2.80378494620370130161308447703, 3.68670878539381830801305511269, 4.63545890380665990141670986613, 5.53618566531056577967176687482, 5.98771079406763610733343708056, 6.74373972124873346554358089110, 7.31428750094090372811842113001